His last publication, which appeared in 1878, was on spherical harmonics (Beitreige zur Theorie der Kugelf unctionen) .
For the subjects of this heading see the articles DIFFERENTIAL EQUATIONS; FOURIER'S SERIES; CONTINUED FRACTIONS; FUNCTION; FUNCTION OF REAL VARIABLES; FUNCTION COMPLEX; GROUPS, THEORY OF; INFINITESIMAL CALCULUS; MAXIMA AND MINIMA; SERIES; SPHERICAL HARMONICS; TRIGONOMETRY; VARIATIONS, CALCULUS OF.
Gauss in particular employed it in the calculation of the magnetic potential of the earth, and it received new light from Clerk Maxwell's interpretation of harmonics with reference to poles on the sphere.
Increased pressure of breath and tension of lips, harmonics could be obtained.
Some curious effects were observed in the formation of harmonics in the rear of the primary tone used.
The successive terms of (21) are called the harmonics of the first term.
It may, however, be stated here that certain experiments of Helmholtz appear to show that the epoch of the harmonics has not much effect on the quality.
Now we may resolve these trains by Fourier's theorem into harmonics of wave-lengths X, 2X, 3A, &c., where X=2AB and the conditions as to the values of y can be shown to require that the harmonics shall all have nodes, coinciding with the nodes of the fundamental curve.
If then we resolve Ahbkc into harmonics by Fourier's theorem, we may follow the motion of the separate harmonics, and their superposition will give the form of the string at any instant.
Further, the same harmonics with the same amplitude will always be present.
But the harmonics are most readily heard if we fortify the ear by an air cavity with a natural period equal to that of the harmonic to be sought.
The whole series of fundamental and overtones gives the complete set of harmonics of frequencies proportional to 1, 2, 3, 4, ..., and wave-lengths proportional to 1, 2, 3, 4 ..
The complete series of harmonics FIG.
Each of the first few harmonics may be easily obtained by touching the string at the first node of the harmonic required, and bowing at the first loop, and the presence of the nodes and loops may be verified by putting light paper riders of shape A on the string at the nodes and loops.
Since the quality of the note sounded depends on the mixture of harmonics, the quality therefore is to some extent dependent on the point of excitation.
Thus, suppose a fundamental 256 has present with it overtone harmonics 512, 768, 1024, 1280, &c., and that we sound with it the major seventh with fundamental 480, and having harmonics 960, 1440, &c. The two sets may be arranged thus c 256 512 768 1024 1280 h 480 960 1440, and we see that the fundamental of the second will beat 32 times per second with the first overtone of the first, giving dissonance.
See also Spherical Harmonics.) Geodesy.
The third volume includes, however, some theological treatises, and the first part of it is occupied with editions of treatises on harmonics and other works of Greek geometers, some of them first editions from the MSS., and in general with Latin versions and notes (Ptolemy, Porphyrius, Briennius, Archimedes, Eutocius, Aristarchus and Pappus).
The remarkable connexion between Bessel's functions and spherical harmonics was established in 1868 by F.
For the mathematical investigation see Spherical Harmonics and for tables see Table, Mathematical.
Aristotle also formed his Peripatetic school into a kind of college with common meals under a president (6tpxcov) changing every ten days; while the philosopher himself delivered lectures, in which his practice, as his pupil Aristoxenus tells us (Harmonics, ii.
Aristoxenus, at the beginning of the second book of the Harmonics,, gives a graphic account of the astonishment caused by these lectures, of Plato, and of their effect on the lectures of Aristotle.
It may be imagined further that, when he established himself at the Academy, his first care was to draw up a scheme of education, including arithmetic, geometry (plane and solid), astronomy, harmonics and dialectic, and that it was not until he had arranged for the carrying out of this programme that he devoted himself to the special functions of professor of philosophy.
He showed, both by analysis and by synthesis, that quality depends on the order, number and intensity of the overtones or harmonics that may, and usually do, enter into the structure of a musical tone.
The long vault has a certain keynote of its own, which, when firmly struck, excites harmonics, including tones of incredible depth and sweetness.
To how many sad hearts did he come like an angel, with the rich tones of his voice waking harmonics of hope, where before there had been despair and silence?
Not only may the fundamental and its harmonics be obtained separately, but they are also to be heard simultaneously, particularly the earlier ones, which are usually more prominent than those higher in the series.
V., on public buildings, has a preface on the theories of Pythagoras, &c. Its twelve chapters treat - (t) of fora and basilicae, with a description of his own basilica at Fanum; (2) of the adjuncts of a forum (aerarium, prison and curia); (3) of theatres, their site and construction; (4) of laws of harmonics; (5) of the arrangement of tuned bronze vases in theatres for acoustic purposes; (6) of Roman theatres; (7) of Greek theatres; (8) of the selection of sites of theatres according to acoustic principles; (9) of porticus and covered walks; (to) of baths, their floors, hypocausts, the construction and use of various parts; (ii) of palaestrae, xysti and other Greek buildings for the exercise of athletes; (12) of harbours and quays.